Theorem elmpps 35558

Description: Definition of a provable pre-statement, essentially just a reorganization of the arguments of df-mcls . (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mppsval.p	⊢ 𝑃 = (mPreSt‘𝑇)
mppsval.j	⊢ 𝐽 = (mPPSt‘𝑇)
mppsval.c	⊢ 𝐶 = (mCls‘𝑇)

Assertion

Ref	Expression
elmpps	⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝐽 ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻)))

Proof of Theorem elmpps

Dummy variables 𝑎 𝑑 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ot 4640	. . 3 ⊢ ⟨𝐷, 𝐻, 𝐴⟩ = ⟨⟨𝐷, 𝐻⟩, 𝐴⟩
2		mppsval.p	. . . 4 ⊢ 𝑃 = (mPreSt‘𝑇)
3		mppsval.j	. . . 4 ⊢ 𝐽 = (mPPSt‘𝑇)
4		mppsval.c	. . . 4 ⊢ 𝐶 = (mCls‘𝑇)
5	2, 3, 4	mppsval 35557	. . 3 ⊢ 𝐽 = {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))}
6	1, 5	eleq12i 2832	. 2 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝐽 ↔ ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))})
7		oprabss 7540	. . . 4 ⊢ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ⊆ ((V × V) × V)
8	7	sseli 3991	. . 3 ⊢ (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} → ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V))
9	2	mpstssv 35524	. . . . . 6 ⊢ 𝑃 ⊆ ((V × V) × V)
10	9	sseli 3991	. . . . 5 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → ⟨𝐷, 𝐻, 𝐴⟩ ∈ ((V × V) × V))
11	1, 10	eqeltrrid 2844	. . . 4 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V))
12	11	adantr 480	. . 3 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻)) → ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V))
13		opelxp 5725	. . . 4 ⊢ (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V) ↔ (⟨𝐷, 𝐻⟩ ∈ (V × V) ∧ 𝐴 ∈ V))
14		opelxp 5725	. . . . 5 ⊢ (⟨𝐷, 𝐻⟩ ∈ (V × V) ↔ (𝐷 ∈ V ∧ 𝐻 ∈ V))
15		simp1 1135	. . . . . . . . . 10 ⊢ ((𝑑 = 𝐷 ∧ ℎ = 𝐻 ∧ 𝑎 = 𝐴) → 𝑑 = 𝐷)
16		simp2 1136	. . . . . . . . . 10 ⊢ ((𝑑 = 𝐷 ∧ ℎ = 𝐻 ∧ 𝑎 = 𝐴) → ℎ = 𝐻)
17		simp3 1137	. . . . . . . . . 10 ⊢ ((𝑑 = 𝐷 ∧ ℎ = 𝐻 ∧ 𝑎 = 𝐴) → 𝑎 = 𝐴)
18	15, 16, 17	oteq123d 4893	. . . . . . . . 9 ⊢ ((𝑑 = 𝐷 ∧ ℎ = 𝐻 ∧ 𝑎 = 𝐴) → ⟨𝑑, ℎ, 𝑎⟩ = ⟨𝐷, 𝐻, 𝐴⟩)
19	18	eleq1d 2824	. . . . . . . 8 ⊢ ((𝑑 = 𝐷 ∧ ℎ = 𝐻 ∧ 𝑎 = 𝐴) → (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ↔ ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃))
20	15, 16	oveq12d 7449	. . . . . . . . 9 ⊢ ((𝑑 = 𝐷 ∧ ℎ = 𝐻 ∧ 𝑎 = 𝐴) → (𝑑𝐶ℎ) = (𝐷𝐶𝐻))
21	17, 20	eleq12d 2833	. . . . . . . 8 ⊢ ((𝑑 = 𝐷 ∧ ℎ = 𝐻 ∧ 𝑎 = 𝐴) → (𝑎 ∈ (𝑑𝐶ℎ) ↔ 𝐴 ∈ (𝐷𝐶𝐻)))
22	19, 21	anbi12d 632	. . . . . . 7 ⊢ ((𝑑 = 𝐷 ∧ ℎ = 𝐻 ∧ 𝑎 = 𝐴) → ((⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ)) ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻))))
23	22	eloprabga 7541	. . . . . 6 ⊢ ((𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V) → (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻))))
24	23	3expa 1117	. . . . 5 ⊢ (((𝐷 ∈ V ∧ 𝐻 ∈ V) ∧ 𝐴 ∈ V) → (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻))))
25	14, 24	sylanb 581	. . . 4 ⊢ ((⟨𝐷, 𝐻⟩ ∈ (V × V) ∧ 𝐴 ∈ V) → (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻))))
26	13, 25	sylbi 217	. . 3 ⊢ (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V) → (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻))))
27	8, 12, 26	pm5.21nii 378	. 2 ⊢ (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻)))
28	6, 27	bitri 275	1 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝐽 ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻)))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  Vcvv 3478  ⟨cop 4637  ⟨cotp 4639   × cxp 5687  ‘cfv 6563  (class class class)co 7431  {coprab 7432  mPreStcmpst 35458  mClscmcls 35462  mPPStcmpps 35463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-ot 4640  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpst 35478  df-mpps 35483

This theorem is referenced by:  mthmpps  35567  mclspps  35569